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Homotopy theory of the master equation package applied to algebra and geometry: a sketch of two interlocking programs

Dennis Sullivan — 2009

Banach Center Publications

Using the algebraic theory of homotopies between maps of dga's we obtain a homotopy theory for algebraic structures defined by collections of multiplications and comultiplications. This is done by expressing these structures and resolved versions of them in terms of dga maps. This same homotopy theory of dga maps applies to extract invariants beyond homological periods from systems of moduli spaces that determine systems of chains that satisfy master equations like dX + X*X = 0. Minimal models of...

A formula for topology/deformations and its significance

Ruth LawrenceDennis Sullivan — 2014

Fundamenta Mathematicae

The formula is e = ( a d e ) b + i = 0 ( B i ) / i ! ( a d e ) i ( b - a ) , with ∂a + 1/2 [a,a] = 0 and ∂b + 1/2 [b,b] = 0, where a, b and e in degrees -1, -1 and 0 are the free generators of a completed free graded Lie algebra L[a,b,e]. The coefficients are defined by x / ( e x - 1 ) = n = 0 B / n ! x . The theorem is that ∙ this formula for ∂ on generators extends to a derivation of square zero on L[a,b,e]; ∙ the formula for ∂e is unique satisfying the first property, once given the formulae for ∂a and ∂b, along with the condition that the “flow” generated by e moves a to b in unit...

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